等级反应模型下多级评分形式的难度等级作用探讨

The Weighted Function Effect of the Polytomous Score and Difficulty Rank Under Graded Response Model

在等级反应模型(GRM)下成倍增加多级评分试题的分数等级,项目信息量只是小幅度地增加,而不是成倍地增加。从能力估计的角度分析,当难度b大于能力估计值θ,随着难度参数增大,被试能力步长也增大,但增大的幅度逐渐变缓,当b-θ>4时能力步长几乎没有变化;当难度b小于能力估计值θ,随着难度参数减小,被试后退的能力步长的绝对值也增大,但增大的幅度逐渐变缓,当b-θ<-4时能力步长绝对值的增大幅度几乎没有变化,即能力步长后退幅度没有变化,存在极限。GRM下使用被试得分进行加权,当能力估计值θ小于难度b时,使用被试得分加权会造成被试估计值成比例地减小,即被试得分分值越大,但被试能力估计值越小,在逻辑上出现矛盾。因此,从项目信息量、被试能力估计角度的分析结果来看,GRM难以有效实现多级评分的算术加权作用。

Under the ideal normal distribution,the difficulty parameters of the two-parameter logistic model and the ratio of the item scores exist in the equality relations.So the difficulty parameters of the Samejima’s graded response model(GRM)and the ratio of the item scores also exist in the equality relations.In the actual test,when the difficulty parameters of the items are estimations via Marginal Maximum Likelihood estimation and EM algorithm,the value of the difficulty parameters depends on the ratio of the item score.Under the GRM,even if the categories of the polytomous item are increased by a wide margin,the amount of item information will increase only on a small scale.In an ideal polytomous test,when the difficulty parameter of the trial item is larger than the ability estimation,the ability step rises with the increasing of difficulty parameters.However,when the distance between the difficulty parameter and the ability estimation is greater than 4,the ability step will not increase and the ability step limit exits.On the other hand,when the difficulty parameter is less than the ability estimation theta,as the difficulty parameter decrease,the absolute value of the ability step increase,which means that the the ability decrease seriously becomes more serious.When the distance between the difficulty parameter and the ability is less than-4,the absolute value of the ability step back does not change,which means the ability step limit exits.Under the GRM,the polytomous score is the difficulty category,not the real weighted score.When the score of the subjects is used to weight the estimated ability in the test,and the ability is less than the difficulty parameter,the subject score weighting will reduce the estimated ability seriously.Under GRM,the multi-level scoring can only reflect the level of difficulty parameters,but can’t effectively reflect the arithmetic weighting function of the scores.